Theorem 1
Tag 0AH3, permanent. Cite it as [Bourbaki, Tag 0AH3].
Lie Groups and Lie Algebras, chapter III, § 7. Lie groups over an ultrametric field
Let L be a complete normable Lie algebra. There exists a Lie group G such that L(G) is isomorphic to L. Two such groups are locally isomorphic.
The first assertion has been proved in § 4, no. 2, Lemma 3. The second is a special case of § 4, no. 2, Theorem 2.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.